Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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82 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThen, if X 1-S are sampled randomly from [a,b] and y rs from [m,M], then the integral canbe estimated byI = (M - m)(b - a) ^Nif N is the number of experiments and n is the number of cases where the conditionv, « fix,)is fulfilled.The proof of this theorem is straightforward from the geometrical interpretation of theintegrals, the rigorous proof is analogous to that of Theorem 2.4.Let us note that the theorem concerns only nonnegative functions. Now in case theintegral of a function assuming also negative values is to be estimated the function is to beformally written in the formf(x) = I +(X) - f (X)wheref +(x) = fix), if f(x) > 0--- 0, otherwiseandMx) = -f(x), if f(x) < 0= 0, otherwiseNow Theorem 4,1 applies to both 12. (x) and f._(x) and the difference of the two estimatesdetermines the estimate of the original function. A similar trick can also be applied in allthe integration methods below, although, it is only of theoretical interest in most applicationsin this book since particle transport calculations almost exclusively assume positive valuedintegrands.The second method is based on the calculation of averages of randomly selected valuesof the integrand.Theorem 4.2. — Let us rewrite Equation (4.1) in the formdxf(x) - JdXh(X) 1P(X) (4.2)where cp(x) 3 0, fdxcp(x) = 1, i.e., cp(x) is a PDF. If N independent X 1samples are selectedfrom tp(x), then the average of the h(x,) weights is an unbiased estimator of I:I =NS "(X 1) (4.3)
83Proof. The expected value of I of Equation (4.3) is(I) = ~ S IdX 1I 1(X 1)CP(X 1) = ^ 2 |d X if( X i) = IN J N .... i JB. GENERALIZATION TO MULTIDIMENSIONAL CASESGeneralization of the method above to multi-dimensional integrals is obvious It isdemonstrated through the example of the second basic method — the one which is preferredin practical applications. Let the integral to be estimated have the following form:dxf(x) (4,4)where x is the symbol of a point in an n-dimensional phase space, and T is the domainintegration. Let us rewrite Equation (4.4) similarly to (4.2):^dXh(X)(P(X) (4.5)wherecp(x) 3= 0, JDX
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Library of Congress Cataloging-in-P
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III. Statistical Considerations •
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IV. Further Generalizations — 183
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Appendix 6A:Unbiased Estimation of
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2 Monte Carlo Particle Transport Me
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Chapter 2INTRODUCTIONWhen we starte
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thus,As it is proved" — and is so
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9B. THE PROBABILITY MIXING METHODTh
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11It is easy to prove 15by summing
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13but the most probable values:x, =
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ISand give a random sign (by the us
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18 Monte Carlo Particle Transport M
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25!) Monte Carlo Particle Transport
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305Chapter 6SPECIAL GAMESIn the maj
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3©7A. CORRELATED MOMENT EQUATIONSI
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309Equation (6.1) will be referred
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311The second moment of the score d
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313andC S(P',P")/C S(P',P") = y(P',
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315introduced in Equation (6.1) for
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317andL„(P 0,P;,...,Pi + 1) = L n
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319whereB„(P„,P; P:, ,) = A*(P
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E. EXAMPLES AND SPECIAL TECHNIQUESL
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323The weight factors associated wi
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325It is seen from Equation (6.37)
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327G. PARAMETRIC PERTURBATIONS: INT
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129Hall 31 gave a constructive deri
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331kernels, which remain unchanged
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333and from Equation (6.51)w';,,(i)
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335According to Equations (6,55) th
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337Minimization is performed by set
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339and letdsdaLet us consider a cor
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341whereis the length of the flight
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and from Equation (6.79)I / ds \ 2
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345Let us assume that we are intere
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347Introduction of this factor also
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3492. Let i|/ n (P) denote the solu
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3SJNow, if ( "> > 0, then k = i an
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353andk„ = S l "VS'" •" (6. if;
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3SSLet k„ denote the normalizatio
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3572. It is then proved that with t
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and start new histories until N new
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361The application of source iterat
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363we haveD 2 [k] = < 2 (k, - k,) )
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365neutron density, S 0 for every
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367if the same relation holds for t
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369Then the multiplication factors
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371in the unperturbed system, the w
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373The perturbation of the effectiv
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375where P" = (r',E'). Multiplying
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377iterate of the derivative of the
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579where P* = (r*,w*,E) = (r*,E*).
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381This estimator, unlike f, in Equ
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383tends to some stable attracting
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3§SIo the case of the next-event e
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387LDFIGURE 6.1.Geometry in scoring
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389Let us denoteThen Equation (6.18
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391is to be increased in order to o
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393Now, since bl(P) = g(P) is bound
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395is scored at every collision wit
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397and its expectation isR — R 1n
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3994. If O > 0 m, then the particle
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401plays a distinguished role), and
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403Theorem 6.7 — If x and s denot
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this a priori information, the best
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40?Thus, the bias introduced into t
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409In view of Equation (6.226) and
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41.1This means that if we use the k
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413Then the whole-sample and rare-s
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41.5According to the results of Sec
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41?(Note that vD 2 [|i|v] is indepe
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419withkOn the other hand, § is an
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421Thus, the expected ratio reads
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423St follows from Equations (6.265
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425Wc start from the observation th
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427Nowi'6.77K)kj„,and therefore(V
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42*3Similarly, for the fourth momen
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43 sOn the other hand, multiplying
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43.¾This will be proven in the lem
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435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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6. Brown. F. B. and Martin, W. R.,
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67. Rief, H. and Fioretti, A., Mont
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Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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48© Monte Carlo Particle Transport
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484 Monte Carlo Panicle Transport M
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486 Monte Carlo Panicle Transport M
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a ( X506 Monte Carlo Particle Trans
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513INDEXAAbsorptiondefined, 25photo
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515score probability in general tim
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517path stretching, 447—455splitt
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